ClearAll[f, fc]
f::zero = "x is zero.";

(* compiled part *)
fc =
  Compile[{{x, _Complex}},
   (* First part of the code; can be anything 
      .
      .
   *)
   If[x == 0, (* Error test*)
    {0, 0}, (* Error response, first part is status, second part is dummy *)
    {1, x^2} (* Normal response, first part is status, second part actual result *)
   ]
];

(* wrapper part *)
f[x_] := With[{res = fc[x]}, 
  If[Chop[res[[1]]] == 0, (* check status *)
    Message[f::zero]; $Failed, (* error *)
    res[[2]] (* normal result *)
  ]]

f[0]

f[1]
(* 1. + 0. I *)

ClearAll[f, fc]
f::zero = "x is zero.";

(* compiled part *)
fc =
  Compile[{{x, _Complex}},
   (* First part of the code; can be anything 
      .
      .
   *)
   If[x == 0, (* Error test*)
    {0, 0}, (* Error response, first part is status, second part is dummy *)
    {1, x^2} (* Normal response, first part is status, second part actual result *)
   ]
];

(* wrapper part *)
f[x_] := With[{res = fc[x]}, 
  If[Chop[res[[1]]] == 0, (* check status *)
    Message[f::zero]; $Failed, (* error *)
    res[[2]] (* normal result *)
  ]]

f[0]

f[1]
(* 1. + 0. I *)

or with MichaelE2's suggestion:

ClearAll[f, fc]
f::zero = "x is zero.";

fc =
  Compile[{{x, _Complex}},
   (* First part of the code *)
   If[x == 0, (* Error test*)
    {},(* Error response, empty list *)
    {x^2} (* Normal response, list with result *)
    ]
   ];
f[x_] := With[{res = fc[x]}, If[res == {}, Message[f::zero]; $Failed, First@res]]